When I try using an @State property wrapper in my first struct, FlagImage, it doesn't work. Xcode gives me this error: "'FlagImage' initializer is inaccessible due to 'private' protection level" - Why is it doing this? I don't really understand the error.

I took @State off and it works just fine. Below is my code of the app that works without the property wrapper

import SwiftUI

struct FlagImage: View {
    var flag: String

    var body: some View {
        Image(flag)
            .renderingMode(.original)
            .clipShape(Capsule())
            .shadow(radius: 8)
    }
}

struct Title: ViewModifier {
    func body(content: Content) -> some View {
        content
            .font(.largeTitle)
            .foregroundColor(.accentColor)
            .padding()
    }
}

extension View {
    func titleStyle() -> some View {
        modifier(Title())
    }
}

struct ContentView: View {

    @State private var showingScore = false
    @State private var scoreTitle = ""
    @State private var userScore = 0
    @State private var maxNumberOfQuestions = 8

    @State private var questionTitle = ""
    @State private var endGame = false
    @State private var currentQuestion = 1
    @State private var correctAnswer = Int.random(in: 0...2)

    @State private var countries = ["Estonia", "France", "Germany", "Ireland", "Italy", "Nigeria", "Poland", "Russia", "Spain", "UK", "US"].shuffled()

    var body: some View {
        ZStack {

            RadialGradient(gradient: Gradient(colors: [Color("Light Pink"), Color("Light Blue")]), center: .bottom, startRadius: 200, endRadius: 400)
                .ignoresSafeArea()

            VStack {

                Spacer()

                HStack {

                    Image(systemName: "flag.filled.and.flag.crossed").foregroundColor(.black)
                    Text("Guess the Flag")
                        .titleStyle()
                    Image(systemName: "flag.and.flag.filled.crossed").foregroundColor(.black)
                }

                VStack(spacing: 15) {

                    Text("Tap the flag of")
                        .foregroundStyle(.secondary)
                        .font(.subheadline.weight(.heavy))
                    Text(countries [correctAnswer]).font(.largeTitle.weight(.semibold))

                    ForEach(0..<3) { number in
                        Button {
                            print("tapped \(countries[number])")
                            flagTapped(number)

                        } label: {
                            FlagImage(flag: countries[number])
                        }
                    }
                }
                .frame(maxWidth:350)
                .padding(.vertical, 20)
                .background(.thinMaterial)
                .clipShape(RoundedRectangle(cornerRadius: 100))

                Spacer()
                Spacer()

                Text("Score: \(userScore)")
                    .foregroundColor(.black)
                    .font(.title.bold())
                Text("Question: \(currentQuestion)")
                    .foregroundColor(.black)
                    .font(.title.bold())

                Spacer()
            }
            .padding()
        }
        .alert(questionTitle, isPresented: $endGame) {
            Button("RESTART GAME", action: newGame)
        } message: {
            Text("Congrats! Final Score: \(userScore).")
        }
        .alert(scoreTitle, isPresented: $showingScore) {
            Button("Continue", action: askQuestion)
        } message: {
            Text("Your Score: \(userScore).")
        }
    }
    func flagTapped(_ number: Int) {
        if number == correctAnswer {
            scoreTitle = "Correct"
            userScore += 10
        } else {
            scoreTitle = "Wrong! It's the flag of \(countries[correctAnswer])."
        }
        showingScore.toggle()
    }
    func askQuestion() {
        if currentQuestion < 8 {
            countries.shuffle()
            currentQuestion += 1
            correctAnswer = Int.random(in: 0...2)
        } else {
            endGame.toggle()
        }
    }
    func newGame() {
        if currentQuestion == 8 {
            userScore = 0
            currentQuestion = 0
            countries.shuffle()
        }
    }
}

struct ContentView_Previews: PreviewProvider {
    static var previews: some View {
        ContentView()
    }
}

I ran your project by copy and pasting to a new project and got no errors!

As @NigelGee says, your code works. As a general rule, when asking for help you should post the non-working code for people to play around with.

Anyway, presumably your non-working code had something like this:

@State private var flag: String

That would make this line:

FlagImage(flag: countries[number])

have the exact error you reference in your post. That's because flag is marked as private.

You can fix the error by adding an initializer to FlagImage like so:

    init(flag: String) {
        self._flag = State(wrappedValue: flag)
        //this initializes the @State property wrapper with the passed-in value
    }

But as you've found, you can also fix that error by removing the @State private part. In this case, @State is not actually necessary since you don't make any changes to flag inside the FlagImage struct. @State is used when you need to change a property of a View; if you aren't changing a property, it doesn't need to be marked as @State.